Difference of sets and power set question

Question

I am asked to state whether the following statements are TRUE or FALSE

$\emptyset$ is in $P(A)$.

$\emptyset$ is subset of A.

My answer is that both of these statements are true.

Answer 1

Your reasoning is perfect. To expand a little bit:

$\varnothing \subseteq B$, so $\varnothing \in P(B)$, so $\varnothing \notin P(A) \setminus P(B)$.

$\varnothing \in A$, so $\{\varnothing\} \subseteq A$, so $\{\varnothing\} \in P(A)$. At the same time, $\varnothing \notin B$, so $\{\varnothing\} \nsubseteq B$, so $\{\varnothing\} \notin P(B)$. Thus, $\{\varnothing\} \in P(A) \setminus P(B)$.

Answer 2

You are given that

$A=\{\phi,\{\phi\}\}$ $\Rightarrow P(A)=\{\phi, \{\phi\},\{\{\phi\}\}, \{\phi,\{\phi\}\}\}$

Also $B=\{\{\phi\}\}$ $\Rightarrow P(B)=\{\phi,\{\phi\}\}$

Now $P(A)\P(B)=\{\{\{\phi\}\}, \{\phi,\{\phi\}\}\}$ and

$P(B)\P(A)=\{\}=\phi$

Hope now you can decide about the asked what is true in that and what is false.